The aim of a drug eluting stent is to prevent restenosis of arteries following percutaneous balloon angioplasty. A long term goal of research in this area is to use modelling to optimise the design of these stents to maximise their efficiency. A key obstacle to implementing this is the lack of a mathematical model of the biology of restenosis. Here we investigate whether mathematical models of cancer biology can be adapted to model the biology of restenosis and the effect of drug elution. We show that relatively simple, rate kinetic models give a good description of available data of restenosis in animal experiments, and its modification by drug elution.

Figure 1.
Experimental data: neointimal thickness in rabbit iliac artery. Reproduced from [7]. Each data point denotes the mean value across several experiments, with 23 rabbits in total for each the bare-metal and the drug-eluting stents. The lines joining the dots are a linear interpolation. In practice, the behaviour of the intima for times between these data points has not been experimentally determined, and it is a major point of this work to better elucidate the dynamics over the entire post-implantation time.

Figure 2.
The modelled cell cycle is shown. Rates of change from $Q$ to $P$ and back are denoted $\beta$ and $\alpha$ respectively. Loss rates from the two phases are denoted by $\lambda_{P}$ and $\lambda_{Q}$. Finally, the growth rate in the proliferative phase is denoted with $\gamma$.

Figure 3.
Phase plot of proliferative cell fraction. $\frac{\partial P}{\partial t}$ vs. $P$ is shown for both the healthy and inflamed arteries. The distance between these solutions may be used to estimate the net transition rate, $\psi$. This distance is shown on the plot with a double-headed arrow.

Figure 4.
Proliferative fraction, $P$, in response to drug-free stent implantation. Note the presence of two temporal domains. On the first, depicted with a dashed line, there is an inflammatory response to the implantation of a stent (cf. $\bar{P}_{1}$, equation (22)). On the second, indicated with a dotted line, the vasculature is returning to its normal state (cf. $\bar{P}_{2}$, equation (27)).

Figure 5.
The increase in the intimal thickness, $L(t)$, in response to drug-free stent implantation. As with Figure 4, note the presence of two temporal domains. On the first, the increase in thickness corresponding to the inflamed response given in equation (31) is shown with a dashed line. The second corresponds to the return to steady state corresponding to equation (33).